Dynamic Graphing on the fx-9750GIII – Helping Students Discover and See Relationships

We talk a lot about how learning math should NOT be focused on memorizing steps and formulas, but instead on doing and discovering the relationships and building understanding. Ideally, students should be visually seeing multiple representations of concepts (tables, graphs, words, pictures, models, manipulatives, equations, etc), comparing and analyzing these representations, and making their own conjectures and ‘rules’. Instead of telling them what they should see or know or do, they are able to figure it out themselves by focusing on relationships and connections (obviously with good structured investigations and questions and activities). This is the ideal.

Research shows that when students discover the patterns and come up with rules/formulas/relationships themselves, they are much more likely to retain the information, or more to the point, more able to ‘recreate’ the experience they had and recall and/or rebuild the information. Not possible when memorizing isolated facts or skills. It’s why just teaching struggling students rules and steps and skills and making them repeat it over and over does not improve learning…..there are no real connections being made.

Algebra is a subject that many students struggle with because of the ‘unknowns’ or variables – it is abstract, and without exploring patterns and relationships of the variables in many forms, it is hard to help students understand and make sense of things like “equations to model real-world situations”. Sadly, there is still the tendency to just teach process – i.e. follow these steps, or “use the quadratic equation to get the solutions” without even talking about what those solutions mean, where they are located, why there are two (or one or none). If we just teach skills and if students just memorize processes/steps, when confronted with something similar, but not exactly the same, they can’t do it, and give up, because they have not had experiences that help them build understanding and see relationships and connections to prior knowledge.

This leads me to today’s focus – Dynamic Graphing.

Dynamic mathematics is when something in a given representation is changing (a measure, a value, a construct), and as it changes, you see the impact of that change. As an example, in geometry, this might mean a triangles three angles are measured, and as you move one or more of the vertices, the angle measures change as you see the triangle change. If you sum the three angle measures, you get 180, and as you change the triangle dynamically, even though the triangle is now different, with different measures that are changing as the triangle changes, the sum of those angles is still 180. Which leads to a conjecture.

Dynamic graphing is the same thing – it’s using variables (instead of static numbers), in equations and changing one or more of them, and then watching what happens to the graph as that variable changes. Dynamic graphing really emphasizes what a variable means – i.e. a quantity that varies, and as it varies, the graph/tables/equations also change. As an example, in a linear equation, defined as y=Ax+b, if we make A- a dynamic variable and change it, students will see the line change it’s ‘steepness’ and direction, and make a conjecture that A- must have something to do with a line’s steepness. If we change b- dynamically, they will see the line move up and down vertically (but not change it’s direction/steepness) – so they conjecture that b- determines where the line crosses the y-axis (because they notice the line always seems to cross at the y-axis at a point that corresponds to b-). Simple things like dynamically changing and visually seeing the impact of that change can lead students to more deeply understand what those coefficients/variables in an equation mean and do, so that when you give them y-=-3x+5, they already know the direction of the line (negative), how steep it might be, and where it’s going to cross the y- axis. You didn’t have to tell them – they figured it out by observing, looking for patterns, and making conjectures and then, most importantly, having discussion with others to confirm their findings.

I wanted to share how you can do dynamic graphing on the new fx-9750GIII graphing calculator. Not a new feature – it’s possible on all the Casio graphing calculators. However, I don’t think many people realize this functionality exists, especially on a black-and-white, inexpensive (but powerful!) graphing calculator. It’s something I think many people expect only mathematics software to be able to do, so I wanted to show you this feature. (And, you can do dynamic geometry too with the Geometry Add-in Menu!)

Video: fx-9750GIII: Dynamic Graphing and Visualizing Variable Changes

Be sure to visit Casio Cares: https://www.casioeducation.com/remote-learning

Here are quick links:

Trigonometric Music – Representing Audible Notes with Sine Waves

My focus this week has been on music and math. Today’s lesson we are going to look at representing musical notes using sine waves. I’ve adapted an activity from Fostering Mathematical Thinking with Music, Casio 2015, that relates frequency (Hertz) of to the sine wave. Also it connects function transformations to the pitch of the frequency and creating a model for the tuning note of A (above the middle C) at 440 Hz using the general form of the sine curve, y=sin(Bx), where x represents time in seconds.

The lesson starts with an interesting little paragraph about oscilloscopes, which I have copied in it’s entirety here, since it relates to the wave pattern of music:Trigonometric Music – Fostering Mathematical Thinking through Music

“An oscilloscope is a piece of scientific lab equipment that draws a graph of an electrical signal. One basic way to use this device is to plug a microphone into the oscilloscope, then produce a sound. The oscilloscope will display a graph of the sound wave on its screen. The investigation (lesson) introduces you to one of the most common wave forms an oscilloscope displays: a sine wave. If you can get your hands on an oscilloscope, try producing many different kinds of sounds, using tuning forks, piano keyboards, various musical instruments, and even the human voice. Even if you produce the same pitch each time, you will fin that the oscilloscope can tell the difference between sounds!”

The Oscilloscope.” evansville.edu. University of Evansville, n.d. Web. 10 Mar. 2014.

The lesson is a really nice connection between trigonometric functions, function transformations and real-world application of mathematics to sound and pitch frequencies. I think students will find all of this fascinating, especially those who play musical instruments. Thinking about music in terms of mathematical sine waves and visualizing how the pitch of the music alters the visualization would be a really great connection, and if you could actually get an oscillator, testing out different sounds would have even more of an impact.

Below is the link to the ClassPad.net version of the activity, along with the original PDF of the activity, which is geared towards the use of a graphing calculator. Now you have both versions and can utilize whichever one makes most sense for your students and resources. The PDF includes sample answers and teaching guidance that can be utilized no matter which version you are using. I’ve also included a video overview that walks through the ClassPad.net version of the activity.

The tool being used in these mini-math lessons is the FREE web-based math software, ClassPad.net.

Remember – if you want to save and/or modify any of these activities, create a free account.  Some useful links below:

Mini-Math Lessons – Middle School Proportional Reasoning (Sampling, Predicting, Arithmetic vs Geometric Sequence)

In today’s lesson focused on proportional reasoning, we are going to look at two different ways to use ratio and proportions to make predictions and comparisons. The first activity is one of my favorite types of activities that I use to do with my students – predicting ‘how many’ based on marking a few, and then taking sample sizes.  This is a great real-world application of proportions, since in the real world, when you hear there are ‘…..x-number of….(name an animal)” left or in this habitat, etc., it’s based on tagging and sampling, NOT really counting. Scientists/conservationists tag animals, record how many are tagged, release them back into their habitat, and then periodically catch the same animals and count the tagged vs. total caught, and use proportions to make an estimate as to how many total animals are in the population in that area. So, it’s an estimate, and the more samples you take, the more accurate your estimate becomes. When I did this with my students, I did it several different ways. How many Starbursts are in a bag? Pull out 10, mark them, put them back in the bag, and then take several samples (returning the samples to the bag each time and shaking it up). Or do something similar with beans, jelly-beans, etc. to represent a population of animals.

Both of these middle-school activities are adapted from Fostering Mathematical Reasoning in the Middle Grades with Casio Technology, Casio 2011. You can find them in the middle school graphing calculator activities as well, where it includes calculator steps. I converted these to ClassPad.net for this post. The first activity today is along those line – i.e. how many fish are in a lake. We use a simulation to simulate catching fish in a net and recording each sample and then making predictions based on individual and cumulative samples. All with proportions. The second activity is using proportions to compare the growth rate of two different plants. It incorporates data collection, scatter plots, graphing, and really comparing and making an argument for what plant is growing fastest based on proportions. It’s a nice activity because it really getting to the heart of making sound mathematical arguments to support the claims. Something students definitely need to be able to do – i.e. defend their thinking and mathematical reasoning and seeing both ‘sides’ of the argument as well.

Here is the link to the two activities and a video overview of both of these.

  1. Capturing the Population
  2. Growing Tall

The tool being used in these mini-math lessons is the FREE web-based math software, ClassPad.net.

Remember – if you want to save and/or modify any of these activities, create a free account.  Some useful links below:

Bouncing Balls, Calculus & Exponentials – Multiple Representations & Data Exploration

One of the cool things about ClassPad.net is the ability to include Animated PNG files, which allows for the collection of ‘real-time’ data. Ish Zamora (@seemathrun), an amazing teacher and teacher leader, has created a couple of new activities that integrate an animated PNG file of a bouncing ball that is used to collect data, do some statistical analysis, calculus and exponential work. These activities are freely available in ClassPad.net’s library of activities (which you can access via your free ClassPad.net account), but I am sharing them in this post as well so you can start playing!!

In these two activities, the same animated PNG is used, but used differently, showing the diverse capabilities

of ClassPad.net to do whatever math you are interested in, based on topic and student level. (BTW – if you are interested in how to add in

animated PNG’s, check out this how-to on your YouTube channel). In the Bouncing Ball activity, which you see a snippet of to the left here, students move data points to the differing heights of the ball as it is bouncing, and then using the statistical menu, find an exponential regression and discuss what is happening to the height of the ball as it continues to bounce over time. This is an exploration in exponential decay. The how-to video below further details working with this activity.

The second activity, Bouncing Ball Calculus, a snapshot of the bouncing ball for one-rebound is taken, and students explore the graph of this function, as seen in the snippet to the right. Students again collect data and create a regression, and find the rate of change between each data point (derivative). Both activities explore different aspects of the bouncing ball, and students are constantly connecting the mathematics to the context of the situation. Both activities help students see that math is happening around them.

These lessons again, are free to use with your students as is. You can also duplicate them in your own free ClassPad.net and add to them or modify them to fit your own classroom needs. Both are great examples of how you can bring some real-world applications into mathematics learning quickly and easily via animated PNG’s, graphs, statistical analysis, calculations, etc. ClassPad.net really helps with providing multiple ways to represent mathematics and discover mathematical connections.

Here is a how-to specific to these two activities that Ish narrates and explains to help you work with these two activities:

Systems of Equations – Sample Lessons and Resources

For this months lesson feature, I am going to focus on Systems of Equations. I chose this topic because I just did a workshop with Algebra 1 teachers in NJ, and this is where they were in their pacing guide, so I am making an assumption that many algebra teachers might also be focusing on this content as well this time of year. I am using a problem from Fostering Algebraic Thinking with Casio Technology in order to provide a real-world problem-solving experience (and I have the resource), but I have altered the problem so that I can utilize the all-in-one capabilities of Classpad.net (tables, graphs, equations, geometry, text).

The Problem

In 2010, there were approximately 950,000 doctors in the United States, and approximately 350,000 of them were primary care doctors. It was estimated that more than 45,000 new primary care doctors will be needed by 2020, but the number of medical school students entering family practice decreased by more than 25 percent from 2002 to 2007. With laws reforming health care, many more people will be insured in the United States. 

For many reasons, including a growing and aging population, the demand for doctors will likely increase in future years. The number of doctors available is also expected to increase. But, due to the high cost of insurance and the fear of malpractice lawsuits, many have predicted that the increase in the number of practicing doctors will not keep up with the increase in demand for doctors.

The table to the right provides data from a study conducted in the state of Michigan. These data approximate the number of doctors that were or will actually be licensed and practicing in Michigan, called the supply, and the number of doctors that were or will be needed by the people of Michigan, called demand.

The question is, will there be enough doctors to provide all the services? The shortage of doctors is a problem that challenges the entire country, not just Michigan.

The Lesson

A shared paper has been created in ClassPad.net called Systems of Equations Help! Not Enough Doctors, which you can access by clicking on the title. The idea behind this problem is to provide a real-world context where students can use tables, graphs, and equations (along with calculations) to create a system of equations. They can solve these using methods such as substitution, elimination, and graphing. Students will also be practicing how to model with mathematics, applying what they know about relationships and being able to create a system of equations that fits the context of the situation in order to find a reasonable solution.

In the activity, there is obviously some focus first on getting students to really understand the problem and what the numbers represent, and then the idea is to have them look for patterns and relationships as they look for a solution. First in the table, then by looking at a scatter plot of the data, where they again try to determine a solution based on a visual. Continuing to look for trends, they use prior knowledge to recognize linear relationships, create equations that model the data, and then graph those equations to find a more precise solution. Then, as a check, they solve their system of equations algebraically. It’s all about multiple representations and helping students see the connections between all the representations, and depending on whether you want a specific, precise answer or just a generalized answer, you might choose a different representation.

ClassPad.net – Lesson In Action

The video below shows the activity and does a brief walk through of some of the components and what it would be like doing the activity from a student perspective. I am a big believer in the think-pair-share approach, so I would suggest having students do the Notice and Wonder individually first, then pair up, then share so that you can make sure that any misunderstandings about the context, and clarification about the numbers is figured out before students start solving. Then I would suggest small groups for working on the problem itself.

Other System of Equation Activities and/or video links